Bhaskar DasGupta. March 28, 2016

Transcription

1 Node Expansions and Cuts in Gromov-hyperbolic Graphs Bhaskar DasGupta Department of Computer Science University of Illinois at Chicago Chicago, IL 60607, USA March 28, 2016 Joint work with Marek Karpinski (University of Bonn) Nasim Mobasheri (UIC) Farzaneh Yahyanejad Supported by NSF grant IIS Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

2 Outline of talk 1 Introduction and Motivation 2 Basic definitions and notations 3 Effect of δ on Expansions and Cuts in δ-hyperbolic Graphs 4 Algorithmic Applications 5 Conclusion and Future Research Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

3 Introduction Various network measures Graph-theoretical analysis leads to useful insights for many complex systems, such as World-Wide Web social network of jazz musicians metabolic networks protein-protein interaction networks Examples of useful network measures for such analyses degree based, e.g. maximum/minimum/average degree, degree distribution, connectivity based, e.g. clustering coefficient, largest cliques or densest sub-graphs, geodesic based, e.g. diameter, betweenness centrality, other more complex measures Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

4 Introduction Gromov-hyperbolicity as a network measure network measure for this talk Gromov-hyperbolicity measure δ originally proposed by Gromov in 1987 in the context of group theory observed that many results concerning the fundamental group of a Riemann surface hold true in a more general context defined for infinite continuous metric space via properties of geodesics can be related to standard scalar curvature of Hyperbolic manifold adopted to finite graphs using a 4-node condition or equivalently using thin triangles Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

5 Basic definitions and notations Hyperbolicity of real-world networks Are there real-world networks that are hyperbolic? Yes, for example: Preferential attachment networks were shown to be scaled hyperbolic [Jonckheere and Lohsoonthorn, 2004; Jonckheere, Lohsoonthorn and Bonahon, 2007] Networks of high power transceivers in a wireless sensor network were empirically observed to have a tendency to be hyperbolic [Ariaei, Lou, Jonckeere, Krishnamachari and Zuniga, 2008] Communication networks at the IP layer and at other levels were empirically observed to be hyperbolic [Narayan and Saniee, 2011] Extreme congestion at a very limited number of nodes in a very large traffic network was shown to be caused due to hyperbolicity of the network together with minimum length routing [Jonckheerea, Loua, Bonahona and Baryshnikova, 2011] Topology of Internet can be effectively mapped to a hyperbolic space [Bogun, Papadopoulos and Krioukov, 2010] Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

6 Motivation Effect of δ on expansion and cut-size Standard practice to investigate/categorize computational complexities of combinatorial problems in terms of ranges of topological measures: Bounded-degree graphs are known to admit improved approximation as opposed to their arbitrary-degree counter-parts for many graph-theoretic problems. Claw-free graphs are known to admit improved approximation as opposed to general graphs for graph-theoretic problems such as the maximum independent set problem. Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

7 Motivation Effect of δ on expansion and cut-size Standard practice to investigate/categorize computational complexities of combinatorial problems in terms of ranges of topological measures: Bounded-degree graphs are known to admit improved approximation as opposed to their arbitrary-degree counter-parts for many graph-theoretic problems. Claw-free graphs are known to admit improved approximation as opposed to general graphs for graph-theoretic problems such as the maximum independent set problem. Motivation for this paper: Effect of δ on expansion and cut-size What is the effect of δ on expansion and cut-size bounds on graphs? For what asymptotic ranges of values of δ can these bounds be used to obtain improved approximation algorithms for related combinatorial problems? Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

8 Outline of talk 1 Introduction and Motivation 2 Basic definitions and notations 3 Effect of δ on Expansions and Cuts in δ-hyperbolic Graphs 4 Algorithmic Applications 5 Conclusion and Future Research Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

9 Basic definitions and notations Graphs, geodesics and related notations Graphs, geodesics and related notations G = (V,E) connected undirected graph of n 4 nodes u v P path P ( u 0,u 1,...,u k 1,u k =u = v ) between nodes u and v l(p ) length (number of edges) of the path u v P ) P u i u j sub-path (u i,u i+1,...,u j of P between nodes u i and u j u v s d u,v a shortest path between nodes u and v length of a shortest path between nodes u and v u 2 u 4 P u 2 u6 P u 2 u6 is the path P ( ) u 2,u 4,u 5,u 6 u 1 u 3 u 5 u 6 l(p )= = 3 d u2,u 6 d u2,u 6 = 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

10 Basic definitions and notations 4 node condition d u2,u 4 Consider four nodes u 1,u 2,u 3,u 4 and the six shortest paths among pairs of these nodes u 1 d u2,u 3 u 2 u 4 d u1,u 4 d u1,u 2 u 1 u 3 d u3,u 4 d u1,u 2 d u3,u 4 d u1,u 3 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

11 Basic definitions and notations 4 node condition d u2,u 4 Consider four nodes u 1,u 2,u 3,u 4 and the six shortest paths among pairs of these nodes u 1 d u2,u 3 u 2 u 4 d u1,u 4 d u1,u 2 u 1 u 3 d u3,u 4 d u1,u 2 d u3,u 4 Assume, without loss of generality, that d u1,u 4 + d u2,u 3 d u1,u }{{} 3 + d u2,u4 d u1,u }{{} 2 + d u3,u }{{} 4 =L =M =S d u1,u Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

12 Basic definitions and notations 4 node condition d u2,u 4 Consider four nodes u 1,u 2,u 3,u 4 and the six shortest paths among pairs of these nodes u 1 d u2,u 3 u 2 u 4 d u1,u 4 d u1,u 2 u 1 u 3 d u3,u 4 d u1,u 2 d u3,u 4 Assume, without loss of generality, that d u1,u 4 + d u2,u 3 d u1,u }{{} 3 + d u2,u4 d u1,u }{{} 2 + d u3,u }{{} 4 =L =M =S d u1,u Let δ u1,u 2,u 3,u 4 = L M 2 + ( + ) 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

13 Basic definitions and notations 4 node condition d u2,u 4 Consider four nodes u 1,u 2,u 3,u 4 and the six shortest paths among pairs of these nodes u 1 d u2,u 3 u 2 u 4 d u1,u 4 d u1,u 2 u 1 u 3 d u3,u 4 d u1,u 2 d u3,u 4 Assume, without loss of generality, that d u1,u 4 + d u2,u 3 d u1,u }{{} 3 + d u2,u4 d u1,u }{{} 2 + d u3,u }{{} 4 =L =M =S d u1,u Let δ u1,u 2,u 3,u 4 = L M 2 + ( + ) Definition (hyperbolicity of G) { } δ(g) = max δu1,u u 1,u 2,u 3,u 2,u 3,u Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

14 Basic definitions and notations Equivalent definition via geodesic triangles Equivalent definition via geodesic triangles (up to a constant multiplicative factor) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

15 Basic definitions and notations Equivalent definition via geodesic triangles Equivalent definition via geodesic triangles (up to a constant multiplicative factor) shortest path u 1 for every ordered triple of shortest paths s s s u 0 u 1, u 0 u 2, u 1 u 2 u 0 u 2 u 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

16 Basic definitions and notations Equivalent definition via geodesic triangles Equivalent definition via geodesic triangles (up to a constant multiplicative factor) shortest path u 1 for every ordered triple of shortest paths s s s u 0 u 1, u 0 u 2, u 1 u 2 u 0 u 2 u 2 s u 0 u 2 u 2 lies in a δ-neighborhood of s s u 0 u 1 u 1 u 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

17 Basic definitions and notations Equivalent definition via geodesic triangles Equivalent definition via geodesic triangles (up to a constant multiplicative factor) shortest path u 1 for every ordered triple of shortest paths s s s u 0 u 1, u 0 u 2, u 1 u 2 u 0 x u 2 u 2 s u 0 u 2 u 2 lies in a δ-neighborhood of s s u 0 u 1 u 1 u 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

18 Basic definitions and notations Equivalent definition via geodesic triangles Equivalent definition via geodesic triangles (up to a constant multiplicative factor) shortest path u 1 for every ordered triple of shortest paths s s s u 0 u 1, u 0 u 2, u 1 u 2 δ u 0 x u 2 u 2 s u 0 u 2 u 2 lies in a δ-neighborhood of s s u 0 u 1 u 1 u 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

19 Basic definitions and notations Equivalent definition via geodesic triangles Equivalent definition via geodesic triangles (up to a constant multiplicative factor) shortest path u 1 for every ordered triple of shortest paths s s s u 0 u 1, u 0 u 2, u 1 u 2 or u 0 x u 2 u 2 s u 0 u 2 u 2 lies in a δ-neighborhood of s s u 0 u 1 u 1 u 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

20 Basic definitions and notations Equivalent definition via geodesic triangles Equivalent definition via geodesic triangles (up to a constant multiplicative factor) shortest path u 1 for every ordered triple of shortest paths s s s u 0 u 1, u 0 u 2, u 1 u 2 δ u 0 x u 2 u 2 s u 0 u 2 u 2 lies in a δ-neighborhood of s s u 0 u 1 u 1 u 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

21 Basic definitions and notations Hyperbolic graphs Definition ( -hyperbolic graphs) G is -hyperbolic provided δ(g) Definition (Hyperbolic graphs) If is a constant independent of graph parameters, then a -hyperbolic graph is simply called a hyperbolic graph Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

22 Basic definitions and notations Hyperbolic graphs Definition ( -hyperbolic graphs) G is -hyperbolic provided δ(g) Definition (Hyperbolic graphs) If is a constant independent of graph parameters, then a -hyperbolic graph is simply called a hyperbolic graph Example (Hyperbolic and non-hyperbolic graphs) Tree: (G)=0 = 0 hyperbolic graph Chordal (triangulated) graph: (G)=1/2 hyperbolic graph Simple cycle: (G)= n/4 non-hyperbolic graph n= 10 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

23 Basic definitions and notations Computational issues Trivially in O ( n 4) time Computation of δ(g) Floyd Warshall algorithm Compute all-pairs shortest paths O ( n 3) time For each combination u1 u 1,u 2,u 3,u 4, compute δ u1,u 2,u 3,u 4 O ( n 4) time Via (max,min) min) matrix multiplication [Fournier, Ismail and Vigneron, 2015] exactly in O ( n 3.69) ) time 2-approximation in in O ( n 2.69) time Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

24 Outline of talk 1 Introduction and Motivation 2 Basic definitions and notations 3 Effect of δ on Expansions and Cuts in δ-hyperbolic Graphs 4 Algorithmic Applications 5 Conclusion and Future Research Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

25 Effect of δ on Expansions in δ-hyperbolic Graphs Definition of node expansion ratio Definition (Node expansion ratio h(s) (n is number of nodes)) witness for h(s) u 2 u 1 S u 4 u 3 u 7 u 5 u 6 S S n 2 h(s)= (S) S S S= = {u 1,u 2,u 3,u 4 } (S)={u 5,u 6,u 7 } { } h= min h(s) S n 2 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

26 Effect of δ on Expansions in δ-hyperbolic Graphs Nested Family of Witnesses for Node Expansion Theorem (Nested Family of Witnesses for Node Expansion) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

27 Effect of δ on Expansions in δ-hyperbolic Graphs Nested Family of Witnesses for Node Expansion Theorem (Nested Family of Witnesses for Node Expansion) Input: graph G = (V,E) with n nodes and m edges maximum node degree d hyperbolicity δ two node p, q with =d d p,q distance between p and q undirected unweighted Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

28 Effect of δ on Expansions in δ-hyperbolic Graphs Nested Family of Witnesses for Node Expansion Theorem (Nested Family of Witnesses for Node Expansion) Input: graph G = (V,E) with n nodes and m edges maximum node degree d hyperbolicity δ two node p, q with =d d p,q distance between p and q undirected unweighted { } For any constant 0<µ<1, there exists at least t = max µ 56log d, 1 nodes S 1 S 2 S t V subsets of S 1 S 2 S t V, each of at most n 2 nodes, with the following properties: j {1,2,..., t} : ) 8ln h (S ( n ) ( ) j min µ, max 500ln n, µ 2 28δ log 2 (2d) All the subsets can be found in a total of O ( n 3 logn+ mn 2) time Either all the subsets contain node p, or all of them contain node q Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

29 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound 8ln ( ) n ( ) 2 min, max 1 1 µ 500ln n, µ 2 28δ log 2 (2d) n nodes, maximum degree d Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

30 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound 8ln ( ) n ( ) 2 min, max 1 1 µ 500ln n, µ 2 28δ log 2 (2d) log 2 n log 2 d n nodes, maximum degree d diameter log 2 n Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

31 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound 8ln ( ) n ( ) 2 min, max 1 1 µ 500ln n, µ 2 28δ log 2 (2d) log 2 n log 2 d n nodes, maximum degree d diameter log 2 n nodes p, q such that =d p,q = log 2 n log 2 d Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

32 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound ( ) log2 d 1 µ 500log max log 2 n, 2 d log µ 2 n 28δlog 1+µ 2 (2d) 2 log 2 n log 2 d n nodes, maximum degree d diameter log 2 n nodes p, q such that =d p,q = log 2 n log 2 d Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

33 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound ( ) log2 d 1 µ 500log max, 2 d log 2 n log µ 2 n 28δlog 1+µ 2 (2d) 2 First component of the bound O ( 1/log 1 µ n ) for fixed d Ω(1) only when d = Ω(n) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

34 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound max ( log2 d log 2 n ) 1 µ 500log, 2 d 2 log µ 2 n 28δlog 1+µ (2d) 2 Second component of the bound suppose G is hyperbolic of constant maximum degree i.e., δ=o(1) and d = O(1) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

35 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound max ( log2 d log 2 n ) 1 µ 500log, 2 d 2 log µ 2 n 28δlog 1+µ (2d) 2 Second component of the bound suppose G is hyperbolic of constant maximum degree i.e., δ=o(1) and d = O(1) 500log 2 d 2 log µ 2 n 28δlog 1+µ (2d) 2 ( = O 1 2 O(1) logµ n ) ( ) 1 = O polylog(n) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

36 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound max ( log2 d log 2 n ) 1 µ 500log, 2 d 2 log µ 2 n 28δlog 1+µ (2d) 2 Second component of the bound suppose G is hyperbolic but maximum degree d is varying i.e., δ= O(1) and d is variable Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

37 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound max ( log2 d log 2 n ) 1 µ 500log, 2 d 2 log µ 2 n 28δlog 1+µ (2d) 2 Second component of the bound suppose G is hyperbolic but maximum degree d is varying i.e., δ= O(1) and d is variable 500log 2 d 2 log µ 2 n 28δlog 1+µ (2d) 2 ( = O expander logdd 2 O(1)logµ n/log 1+µ d Ω(1) only if d > 2 Ω ) d = O logd polylog(n) ( ) log log n log log log n 1 log 1+µ d Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

38 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound max ( log2 d log 2 n ) 1 µ 500log, 2 d 2 log µ 2 n 28δlog 1+µ (2d) 2 Second component of the bound suppose G is not hyperbolic but had constant maximum degree i.e., d = O(1) and δ is variable Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

39 Effect of δ on Expansions in δ-hyperbolic Graphs Asymptotics of the expansion bound Illustration of asymptotics of the expansion bound ( ) log2 d 1 µ 500log max log 2 n, 2 d log µ 2 n 28δlog 2 1+µ (2d) 2 Second component of the bound suppose G is not hyperbolic but had constant maximum degree i.e., d = O(1) and δ is variable ( ) 500log 2 d 1 = O 2 log µ 2 n 28δ log 1+µ (2d) 2 Ω(1) expander 2 O(1) logµ n δ only if δ=ω ( log µ n ) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

40 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Theorem (Witnesses for Node Expansion with Limited Overlaps) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

41 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Theorem (Witnesses for Node Expansion with Limited Overlaps) Input: graph G = (V,E) with n nodes and m edges maximum node degree d hyperbolicity δ two node p, q with =d d p,q distance between p and q undirected unweighted Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

42 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Theorem (Witnesses for Node Expansion with Limited Overlaps) Input: graph G = (V,E) with n nodes and m edges maximum node degree d hyperbolicity δ two node p, q with =d d p,q distance between p and q undirected unweighted For any constant 0<µ<1 and any integer τ< ( ) 1/µ, there exists 42δ log2 (2d) log 2 (2 ) τ/4 distinct collections of subsets of nodes F 1,F 2,...,F τ/4 2 V such that: Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

43 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Theorem (Witnesses for Node Expansion with Limited Overlaps) Input: graph G = (V,E) with n nodes and m edges maximum node degree d hyperbolicity δ two node p, q with =d d p,q distance between p and q undirected unweighted For any constant 0<µ<1 and any integer τ< ( ) 1/µ, there exists 42δ log2 (2d) log 2 (2 ) τ/4 distinct collections of subsets of nodes F 1,F 2,...,F τ/4 2 V such that: (S) max{ j { { ( ) 1,..., τ 4} S 1 1 µ ( ) ( ( /τ) µ Fj : h ( /τ), 360log 2 n / ( /τ)2 7δ log 2 (2d) )} Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

44 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Theorem (Witnesses for Node Expansion with Limited Overlaps) Input: graph G = (V,E) with n nodes and m edges maximum node degree d hyperbolicity δ two node p, q with =d d p,q distance between p and q undirected unweighted For any constant 0<µ<1 and any integer τ< ( ) 1/µ, there exists 42δ log2 (2d) log 2 (2 ) τ/4 distinct collections of subsets of nodes F 1,F 2,...,F τ/4 2 V such that: (S) max{ j { { ( ) 1,..., τ 4} S 1 1 µ ( ) ( ( /τ) µ )} Fj : h ( /τ), 360log 2 n / ( /τ)2 7δ log 2 (2d) { } Each collection F j has t j = max ( /τ) µ 56log 2 d, 1 subsets V j,1,...,v j,t j that form a } 56log 2 d, 1 nested family, i.e., V j,1 V j,2 V j,t j Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

45 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Theorem (Witnesses for Node Expansion with Limited Overlaps) Input: graph G = (V,E) with n nodes and m edges maximum node degree d hyperbolicity δ two node p, q with =d d p,q distance between p and q undirected unweighted For any constant 0<µ<1 and any integer τ< ( ) 1/µ, there exists 42δ log2 (2d) log 2 (2 ) τ/4 distinct collections of subsets of nodes F 1,F 2,...,F τ/4 2 V such that: (S) max{ j { { ( ) 1,..., τ 4} S 1 1 µ ( ) ( ( /τ) µ )} Fj : h ( /τ), 360log 2 n / ( /τ)2 7δ log 2 (2d) { } Each collection F j has t j = max ( /τ) µ 56log 2 d, 1 subsets V j,1,...,v j,t j that form a } 56log 2 d, 1 nested family, i.e., V j,1 V j,2 V j,t j (limited overlap claim) For every pair of subsets V i,k F i and V j,k F j with i j, either V i,k V j,k = or at least 2τ nodes in each subset do not belong to the other subset Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

46 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Theorem (Witnesses for Node Expansion with Limited Overlaps) Input: graph G = (V,E) with n nodes and m edges maximum node degree d hyperbolicity δ two node p, q with =d d p,q distance between p and q undirected unweighted For any constant 0<µ<1 and any integer τ< ( ) 1/µ, there exists 42δ log2 (2d) log 2 (2 ) τ/4 distinct collections of subsets of nodes F 1,F 2,...,F τ/4 2 V such that: (S) max{ j { { ( ) 1,..., τ 4} S 1 1 µ ( ) ( ( /τ) µ )} Fj : h ( /τ), 360log 2 n / ( /τ)2 7δ log 2 (2d) { } Each collection F j has t j = max ( /τ) µ 56log 2 d, 1 subsets V j,1,...,v j,t j that form a } 56log 2 d, 1 nested family, i.e., V j,1 V j,2 V j,t j (limited overlap claim) For every pair of subsets V i,k F i and V j,k F j with i j, either V i,k V j,k = or at least 2τ nodes in each subset do not belong to the other subset All subsets in each F j can be found in a total of O ( n 3 logn+ mn 2) time Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

47 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

48 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

49 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Set = log 2 n log 2 d Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

50 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Set = log 2 n log 2 d Set τ= 1/2 = ( ) log2 n 1/2 log 2 d Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

51 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Set = log 2 n log 2 d Set τ= 1/2 = This gives: ( ) log2 n 1/2 log 2 d Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

52 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Set = log 2 n log 2 d Set τ= 1/2 = ( ) log2 n 1/2 log 2 d This gives: ( Ω (log2 Ω( n ) ) 1/2 nested families of subsets of nodes Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

53 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Set = log 2 n log 2 d Set τ= 1/2 = ( ) log2 n 1/2 log 2 d This gives: ( Ω (log2 Ω( n ) ) 1/2 nested families of subsets of nodes ( each family has Ω (log2 Ω( n ) ) 1/2 subsets each of maximum node ( ) (1 µ)/2 1 expansion O log 2 n Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

54 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Set = log 2 n log 2 d Set τ= 1/2 = ( ) log2 n 1/2 log 2 d This gives: ( Ω (log2 Ω( n ) ) 1/2 nested families of subsets of nodes ( each family has Ω (log2 Ω( n ) ) 1/2 subsets each of maximum node ( ) (1 µ)/2 1 expansion O log 2 n every pair of subsets from different families Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

55 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Set = log 2 n log 2 d Set τ= 1/2 = ( ) log2 n 1/2 log 2 d This gives: ( Ω (log2 Ω( n ) ) 1/2 nested families of subsets of nodes ( each family has Ω (log2 Ω( n ) ) 1/2 subsets each of maximum node ( ) (1 µ)/2 1 expansion O log 2 n every pair of subsets from different families is disjoint Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

56 Effect of δ on Expansions in δ-hyperbolic Graphs Family of Witnesses for Node Expansion With Limited Mutual Overlaps Illustration of the limited overlap bound Suppose that δ and d are constants Set = log 2 n log 2 d Set τ= 1/2 = ( ) log2 n 1/2 log 2 d This gives: ( Ω (log2 Ω( n ) ) 1/2 nested families of subsets of nodes ( each family has Ω (log2 Ω( n ) ) 1/2 subsets each of maximum node ( ) (1 µ)/2 1 expansion O log 2 n every pair of subsets from different families is disjoint ( or has at least Ω (log2 Ω( n ) ) 1/2 private nodes Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

57 Effect of δ on Cuts in δ-hyperbolic Graphs Definition of s-t cut and size of s-t cut cut edges } E (S, s, t)= {e 1,e 2,e 3,e 4 s-t cut s e 2 e 3 e 1 t u 2 u 2 u 1 S e 4 u 3 } V (S, s, t)= {u 1,u 2,u 3 cut nodes Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

58 Effect of δ on Cuts in δ-hyperbolic Graphs Family of Mutually Disjoint Cuts Lemma (Family of Mutually Disjoint Cuts) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

59 Effect of δ on Cuts in δ-hyperbolic Graphs Family of Mutually Disjoint Cuts Lemma (Family of Mutually Disjoint Cuts) graph G = (V,E) with n nodes and m edges undirected unweighted d is maximum degree of any node except s, t and Input: any node within a distance of 35δ of s hyperbolicity δ two node s, t with d s,t > 48δ+8δlog n distance between s and t is at least logarithmic in n Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

60 Effect of δ on Cuts in δ-hyperbolic Graphs Family of Mutually Disjoint Cuts Lemma (Family of Mutually Disjoint Cuts) graph G = (V,E) with n nodes and m edges undirected unweighted d is maximum degree of any node except s, t and Input: any node within a distance of 35δ of s hyperbolicity δ two node s, t with d s,t > 48δ+8δlog n distance between s and t is at least logarithmic in n there exists a set of d s,t 8δlog n = Ω ( ) d s,t (node and edge) disjoint s-t cuts 50δ Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

61 Effect of δ on Cuts in δ-hyperbolic Graphs Family of Mutually Disjoint Cuts Lemma (Family of Mutually Disjoint Cuts) graph G = (V,E) with n nodes and m edges undirected unweighted d is maximum degree of any node except s, t and Input: any node within a distance of 35δ of s hyperbolicity δ two node s, t with d s,t > 48δ+8δlog n distance between s and t is at least logarithmic in n there exists a set of d s,t 8δlog n = Ω ( ) d s,t (node and edge) disjoint s-t cuts 50δ each such cut has at most d 12δ+1 cut edges Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

62 Outline of talk 1 Introduction and Motivation 2 Basic definitions and notations 3 Effect of δ on Expansions and Cuts in δ-hyperbolic Graphs 4 Algorithmic Applications 5 Conclusion and Future Research Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

63 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Network Design Application: Minimizing Bottleneck Edges [Assadi et al., 2014; Omran, Sack and Zarrabi-Zadeh, 2013; Zheng, Wang, Yang and Yang, 2010] applications in several communication network design problems Problem (Unweighted Uncapacitated Minimum Vulnerability (UUMV)) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

64 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Network Design Application: Minimizing Bottleneck Edges [Assadi et al., 2014; Omran, Sack and Zarrabi-Zadeh, 2013; Zheng, Wang, Yang and Yang, 2010] applications in several communication network design problems Problem (Unweighted Uncapacitated Minimum Vulnerability (UUMV)) Input: graph G = (V,E) with n nodes and m edges two node s, t two positive integers 0<r < κ undirected unweighted Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

65 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Network Design Application: Minimizing Bottleneck Edges [Assadi et al., 2014; Omran, Sack and Zarrabi-Zadeh, 2013; Zheng, Wang, Yang and Yang, 2010] applications in several communication network design problems Problem (Unweighted Uncapacitated Minimum Vulnerability (UUMV)) Input: graph G = (V,E) with n nodes and m edges two node s, t two positive integers 0<r < κ undirected unweighted Definition (shared edge) An edge is shared if it is in more than r paths between s and t Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

66 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Network Design Application: Minimizing Bottleneck Edges [Assadi et al., 2014; Omran, Sack and Zarrabi-Zadeh, 2013; Zheng, Wang, Yang and Yang, 2010] applications in several communication network design problems Problem (Unweighted Uncapacitated Minimum Vulnerability (UUMV)) Input: graph G = (V,E) with n nodes and m edges two node s, t two positive integers 0<r < κ undirected unweighted Definition (shared edge) An edge is shared if it is in more than r paths between s and t Goal find κ paths between s and t minimize number of shared edges Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

67 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Minimizing Bottleneck Edges: Known results UUMV does not admit a 2 log1 ε n -approximation for any constant ε>00 unless NP DTIME ( n log log n) even if r = 1 UUMV admits a κ r+1 -approximation 2 log1 ε n However, no non-trivial approximation of UUMV that depends on m and/or n only is currently known { } For r = 1, UUMV admits a min n 3 4, m 1 2 -approximation Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

68 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Minimizing Bottleneck Edges: Our result Lemma (Approximation of UUMV for δ-hyperbolic graphs) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

69 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Minimizing Bottleneck Edges: Our result Lemma (Approximation of UUMV for δ-hyperbolic graphs) graph G = (V,E) with n nodes and m edges d is maximum degree of any node except s, t and Input: any node within a distance of 35δ of s hyperbolicity δ undirected unweighted Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

70 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Minimizing Bottleneck Edges: Our result Lemma (Approximation of UUMV for δ-hyperbolic graphs) graph G = (V,E) with n nodes and m edges d is maximum degree of any node except s, t and Input: any node within a distance of 35δ of s hyperbolicity δ undirected unweighted UUMV can be approximated within a factor of O ( max { logn, d O(δ)}) Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

71 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Minimizing Bottleneck Edges: Our result Lemma (Approximation of UUMV for δ-hyperbolic graphs) graph G = (V,E) with n nodes and m edges d is maximum degree of any node except s, t and Input: any node within a distance of 35δ of s hyperbolicity δ undirected unweighted UUMV can be approximated within a factor of O ( max { logn, d O(δ)}) Remark ( ) log n Lemma provides improved approximation as long as δ δ=o= o log d Our approximation ratio is independent of the value of κ ( ) log n δ=ω log allows expander graphs for which UUMV is expected to be d harder to approximate Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

72 Algorithmic Applications Network Design Application: Minimizing Bottleneck Edges Minimizing Bottleneck Edges: Our result Lemma (Approximation of UUMV for δ-hyperbolic graphs) graph G = (V,E) with n nodes and m edges d is maximum degree of any node except s, t and Input: any node within a distance of 35δ of s hyperbolicity δ undirected unweighted UUMV can be approximated within a factor of O ( max { logn, d O(δ)}) Proof strategy overview Define a new more general problem: edge hitting set problem for size constrained cuts (EHSSC) Show that UUMV has similar approximability properties as EHSSC Provide approximation algorithm for EHSSC using family of cuts lemma Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

73 Algorithmic Applications Small Set Expansion Problem Small Set Expansion Problem [Gandhi and Kortsarz, 2015; Bansal et al., 2011; Raghavendra and Steurer, 2010; Arora, Barak and Steurer, 2010;... ] application: studying Unique Games Conjecture Problem (Small Set Expansion (SSE)) a case of [Theorem 2.1 of Arora, Barak and Steurer, 2010], rewritten as a problem Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

74 Algorithmic Applications Small Set Expansion Problem Small Set Expansion Problem [Gandhi and Kortsarz, 2015; Bansal et al., 2011; Raghavendra and Steurer, 2010; Arora, Barak and Steurer, 2010;... ] application: studying Unique Games Conjecture Problem (Small Set Expansion (SSE)) a case of [Theorem 2.1 of Arora, Barak and Steurer, 2010], rewritten as a problem Definition ( normalized edge expansion ratio Φ(S)) For a subset of nodes S: Φ(S)= number of cut edges from S to V \S sum of degrees of the nodes in S Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

75 Algorithmic Applications Small Set Expansion Problem Small Set Expansion Problem [Gandhi and Kortsarz, 2015; Bansal et al., 2011; Raghavendra and Steurer, 2010; Arora, Barak and Steurer, 2010;... ] application: studying Unique Games Conjecture Problem (Small Set Expansion (SSE)) a case of [Theorem 2.1 of Arora, Barak and Steurer, 2010], rewritten as a problem Definition ( normalized edge expansion ratio Φ(S)) For a subset of nodes S: Φ(S)= number of cut edges from S to V \S sum of degrees of the nodes in S Input: d-regular graph G = (V,E) with n nodes and m edges undirected unweighted G has subset S of ζn nodes, for some constant 0<ζ< 1 2, such that Φ(S) ε ε for some constant 0<ε 1 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

76 Algorithmic Applications Small Set Expansion Problem Small Set Expansion Problem [Gandhi and Kortsarz, 2015; Bansal et al., 2011; Raghavendra and Steurer, 2010; Arora, Barak and Steurer, 2010;... ] application: studying Unique Games Conjecture Problem (Small Set Expansion (SSE)) a case of [Theorem 2.1 of Arora, Barak and Steurer, 2010], rewritten as a problem Definition ( normalized edge expansion ratio Φ(S)) Input: Goal For a subset of nodes S: Φ(S)= number of cut edges from S to V \S sum of degrees of the nodes in S d-regular graph G = (V,E) with n nodes and m edges undirected unweighted G has subset S of ζn nodes, for some constant 0<ζ< 1 2, such that Φ(S) ε ε for some constant 0<ε 1 Find a subset S of ζn nodes such that Φ(S ) ηε for some universal constant η>00 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

77 Algorithmic Applications Small Set Expansion Problem Summary of what is known about SSE computing a good approximation of SSE seems to be quite hard approximation ratio of algorithm in [Raghavendra, Steurer and Tetali, ( 2010] deteriorates proportional to log 1 ζ) O(1)-approximation in [Bansal et al., 2011] works only if the graph excludes two specific minors [Arora, Barak and Steurer, 2010] provides a O ( 2 c n) time algorithm for some constant c < 1 for SSE Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

78 Algorithmic Applications Small Set Expansion Problem Summary of what is known about SSE computing a good approximation of SSE seems to be quite hard approximation ratio of algorithm in [Raghavendra, Steurer and Tetali, ( 2010] deteriorates proportional to log 1 ζ) O(1)-approximation in [Bansal et al., 2011] works only if the graph excludes two specific minors [Arora, Barak and Steurer, 2010] provides a O ( 2 c n) time algorithm for some constant c < 1 for SSE Our result Lemma polynomial time solution of SSE for δ-hyperbolic graphs when δ is sub-logarithmic and d is sub-linear SSE can be solved in polynomial time provided d and δ satisfy: d 2 log 1 3 ρ n and δ log ρ n for some constant 0<ρ< 1 3 Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

79 Outline of talk 1 Introduction and Motivation 2 Basic definitions and notations 3 Effect of δ on Expansions and Cuts in δ-hyperbolic Graphs 4 Algorithmic Applications 5 Conclusion and Future Research Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

80 Conclusion and Future Research We provided the first known non-trivial bounds on expansions and cut-sizes for graphs as a function of hyperbolicity measure δ We showed how these bounds and their related proof techniques lead to improved algorithms for two related combinatorial problems We hope that these results sill stimulate further research in characterizing the computational complexities of related combinatorial problems over asymptotic ranges of δ Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

81 Conclusion and Future Research We provided the first known non-trivial bounds on expansions and cut-sizes for graphs as a function of hyperbolicity measure δ We showed how these bounds and their related proof techniques lead to improved algorithms for two related combinatorial problems We hope that these results sill stimulate further research in characterizing the computational complexities of related combinatorial problems over asymptotic ranges of δ Improve the bounds in our paper Some future research problems Can we get a polynomial-time solution of Unique Games Conjectire for some asymptotic ranges of δ? Obvious recursive approach encounters a hurdle since hyperbolicity is not a hereditary property, i.e., removal of nodes or edges may change δ sharply Can our bounds on expansions and cut-sizes be used to get an improved approximation for the minimum multicut problem for δ δ=o(logn) =? Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29

82 Final slide Thank you for your attention Questions?? Bhaskar DasGupta (UIC) Expansions and Cuts in hyperbolic graphs March 28, / 29